minimal_surface — 3D `surface` op

• Data kinds: none → mesh (an op determined by its arguments alone — it takes no image or data input)

• Call: import fullseye as fs; fs.ledger.minimal_surface(kind='catenoid', nu=80, nv=120, extent=1.5, scale=1.0) (to call the implementation directly, import render3d; render3d.minimal_surface(kind='catenoid', nu=80, nv=120, extent=1.5, scale=1.0); from the registry, ops3d.get("minimal_surface"))

Usage

A classical minimal surface, from its exact parametrisation.

`catenoid (c cosh(u/c) cos v, c cosh(u/c) sin v, u), helicoid`

`(u cos v, u sin v, c v), enneper`

`(u - u^3/3 + u v^2, v - v^3/3 + v u^2, u^2 - v^2) and scherk`

`z = log(cos y / cos x)` (the doubly periodic surface, over one cell).

★★Why this earns its place — the definition is the gate. A minimal surface

is one whose mean curvature vanishes everywhere, and the existing

`vertex_curvature` op measures exactly that. So the claim "this is a minimal

surface" is checked by an op that knows nothing about how the surface was

built. ★A trap worth naming: the catenoid and the helicoid are *isometric*, so

their Gaussian curvatures agree pointwise — but that is a **necessary, not

sufficient** condition for minimality (a sphere and a plane differ in K, yet

matching K would not make either minimal). Grade on H.

Parameters

----------

kind : str

One of `MINIMAL_SURFACES`.

nu, nv : int

Grid resolution along the two parameters.

extent : float

Half-range of the first parameter (how much of the surface to take).

scale : float

The surface's own scale parameter (`c` for catenoid and helicoid).

Returns a `mesh: (vertices, faces)`.

Raises `ValueError`: unknown kind; a grid below 4x4 or over the cap;

non-positive scale; an extent that reaches the singularity (Scherk's surface

is only defined where `cos x and cos y` share a sign, so an extent at or

beyond `pi/2` is refused rather than silently clipped to infinity).

HALCON: no operator.

References (sample data, literature)

• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).

• Operator provenance and references — the sources of the research/methods this op family came from.

Runnable examples (verified samples that actually call this op)

• minimal_surfaces — py -3.11 examples_3d/minimal_surfaces.py

Ops the type connects to (they accept `mesh` as input)

mesh_to_voxel · mesh_to_points · to_points · fuse_to_voxel · ambient_occlusion · cast_shadow · supersample_mesh · render_beauty

Same category (`surface`)

minimal_surface_bend · gyroid_isosurface · gyroid_solid_mask · curve3d_tube_mesh

*Provenance: render3d.py — 3D operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

minimal_surface — 3D surface op